we can define \(h^*(a,w)h^*_{a}(w).\) 
The existence of the bridge function corresponds to the results of [1]} and [2]}.
For simplicity, in constrast to [1]}, we assume that there is no shared quantity between the proxy variables \(W\)  and \(Z\) .
Following the proofs of the previous works [1]} and [2]}, we aim to solve the integral equation in (REF ).
